Transcript z-Scores and the Normal Curve

z-Scores, the Normal Curve,
& Standard Error of the
Mean
I. z-scores and conversions
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What is a z-score?
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A measure of an observation’s distance from the
mean.
The distance is measured in standard deviation
units.
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If a z-score is zero, it’s on the mean.
If a z-score is positive, it’s above the mean.
If a z-score is negative, it’s below the mean.
If a z-score is 1, it’s 1 SD above the mean.
If a z-score is –2, it’s 2 SDs below the mean.
Computing a z-score
z
X 

X X
or z
SD
Examples of computing z-scores
X
X
X X
SD
z
X X
SD
5
3
2
2
1
6
3
3
2
1.5
5
10
-5
4
-1.25
6
3
3
4
.75
4
8
-4
2
-2
Computing raw scores from z scores
X  z   or
z
X X
SD
SD
X  zSD  X
X
zSD
X
1
2
2
3
5
-2
2
-4
2
-2
.5
4
2
10
12
-1
5
-5
10
5
Example of Computing z scores from raw
scores
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List raw scores (use calculator)
Compute mean
Compute SD
Compute z
Review
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Interpret a z score of 1
M = 10, SD = 2, X = 8. Z =?
M = 8, SD = 1, z = 3. X =?
What is the A (SAT) score for a z score of 1?
Definition
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To move from a raw score to a z score, what
must we know about the raw score
distribution?
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1 mean and standard deviation
2 maximum and minimum
3 median and variance
4 mode and range
Application
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If Judy got a z score of 1.5 on an in-class
exam, what can we say about her score
relative to others who took the exam?
1 it is above average
2 it is average
3 it is below average
4 it is a ‘B’
Normal Curve
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The normal curve is continuous.
N
 ( X   ) 2 / 2 2
Y
e
 2
The formula is:
This formula is not intuitively obvious.
The important thing to note is that there are
only 2 parameters that control the shape of
the curve: σ and μ. These are the
population SD and mean, respectively.
Normal Curve
The shape of the distribution changes with
only two parameters, σ and μ, so if we know
these, we can determine everything else.
Normal Curve
20
16
Frequency
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12
8
4
0
-4
-2
0
Score (X)
2
4
Standard Normal Curve
Standard normal curve has a mean of zero
and an SD of 1.
Standard Normal Curve
Probability (Relative Frequency)
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0 .4
50 Percent
0 .3
34.13 %
0 .2
0 .1
13.59%
2.15%
0 .0
-3
-2
-1
0
1
2
Scores in standard deviations from mu
3
Normal Curve and the z-score
If X is normally distributed, there will be a
correspondence between the standard
normal curve and the
z-score.
Standard Normal Curve
Probability (Relative Frequency)
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0 .4
0 .3
0 .2
-3
-1
1
3
5
7
9
Scores in raw score units
0 .1
0 .0
-3
-2
-1
0
1
2
3
Scores in standard deviations from mu
Normal curve and z-scores
We can use the information from the normal
curve to estimate percentages from z-scores.
Standard Normal Curve
Probability (Relative Frequency)
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0 .4
50 Percent
0 .3
34.13 %
0 .2
0 .1
13.59%
2.15%
0 .0
-3
-2
-1
0
1
2
Scores in standard deviations from mu
3
Test your mastery of z
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If a raw score is 8, the mean is 10 and the
standard deviation is 4, what is the z-score?
1: -1.0
2: -0.5
3: 0.5
4: 2.0
Test your mastery of z and the normal
curve
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If a distribution is normally distributed, about
what percent of the scores fall below +1 SD?
1: 15
2: 50
3: 85
4: 99
Tabled values of the normal to
estimate percentages
Z
Between
mean and
z
Beyond z
Z
Between
mean and
z
Beyond z
0.0
50.00
0.90
31.5
18.41
0.10
3.98
46.02
1.00
34.13
15.87
0.20
7.93
42.07
1.10
36.43
13.57
0.30
11.79
38.21
1.20
38.49
11.51
0.40
15.54
34.46
1.30
40.32
09.68
0.50
19.15
30.85
1.40
41.92
08.08
0.60
22.57
27.43
1.50
43.32
06.68
0.70
25.80
24.20
1.60
44.52
05.48
0.80
28.81
21.19
1.70
45.54
04.46
0.00
Estimating percentages
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What z-score separates the bottom 70
percent from the top 30 percent of scores?
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z= .5
Probability (Relative Frequency)
Standard Normal Curve
0 .4
20%
50%
0 .3
30%
0 .2
z=?
0 .1
z=0
0 .0
-3
-2
-1
0
1
2
3
Scores in standard deviations from mu
Estimating percentages
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What z-score separates the top 10 percent
from the bottom 90 percent?
Standard Normal Curve
Z=1.3
Probability (Relative Frequency)
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0 .4
0 .3
40%
50%
0 .2
z=?
10%
0 .1
z=0
0 .0
-3
-2
-1
0
1
2
3
Scores in standard deviations from mu
Percentile Ranks
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A percentile rank is the percentage of cases
up to and including the one in which we are
interested. From the bottom up to the current
score.
Q: What is the percentile rank of an SAT
score of 600?
Percentile Rank
A: First we find the z score [(600500)/100]=1. Then we find the area for z=1.
Between mean and z = 34.13. Below mean
=50, so total below is 50+34.13 or about 84
Standard Normal Curve
percent.
Probability (Relative Frequency)
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0 .4
0 .3
200
0 .2
0 .1
300
400
500
600
700
800
SAT Scores
50%
34.13%
0 .0
-3
-2
-1
0
1
2
3
Scores in standard deviations from mu
Estimating percentages
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Suppose our basketball coach wants to
estimate how many entering freshmen will be
over 6’6” (78 inches) tall. Suppose the
mean height of entering freshmen is 68
inches and the SD of height is 6.67 inches
and there will be 1,000 entering freshmen.
How many are expected to be bigger than 78
inches?
Estimating percentages
Find z, then percent, then the number. Z=(7868)/6.67=1.499=1.5. Beyond z is 6.68
percent. If 100 people, would be 6.68
expected, if 1000, 66.8
or 67 folks.
Standard Normal Curve
Probability (Relative Frequency)
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0 .4
0 .3
54.66 61.33
50%
68
Height
74.67 81.34
?%
0 .2
z=1.5
?%
0 .1
z=0
0 .0
-3
-2
-1
0
1
2
3
Scores in standard deviations from mu
Review
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What z score separates the top 20 percent
from the bottom 80 percent?
What is a percentile rank?
Suppose you want to estimate the
percentage of women taller than the height of
the average man. Say Mmale = 69 in. Mfemale
= 66 in. SDfemale= 2 in. Pct?
Z = (69-66)/2 = 3/2 = 1.5
Beyond z = 1.5 is 6.68 pct.